Linear combinations of Rademacher random variables

For a fixed unit vector \(a=(a_1,a_2,\ldots,a_n)\in S^{n-1}\), we consider the \(2^n\) sign vectors \(\varepsilon=(\varepsilon^1,\varepsilon^2,\ldots,\varepsilon^n)\in \{+1,-1\}^n\) and the corresponding scalar products \(\varepsilon\cdot a=\sum_{i=1}^n \varepsilon^ia_i\). In this paper we will solve for \(n=1,2,\ldots,9\) an old conjecture stating that of the \(2^n\) sums of the form \(\sum\pm a_i\) it is impossible that there are more with \(|\sum_{i=1}^n \pm a_i|>1\) than there are with \(|\sum_{i=1}^n \pm a_i|\leq1\). Although the problem has been solved completely in case the \(a_i\)'s are equal, the more general problem with possible non-equal \(a_i\)'s remains open for values of \(n\geq 10\). The present method can also be used for \(n\geq 10\), but unfortunately the technical difficulties seem to grow exponentially with \(n\) and no "induction type of argument" has been found. The conjecture has an appealing reformulation in probability theory and in geometry. In probability theory the results lead to upper bounds which are much better than for instance Chebyshevnequalities.